Questions in category: 黎曼几何 (Riemannian Geometry)

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11. $n>2$ 维黎曼流形成为 Einstein 流形的充要条件

Posted by haifeng on 2014-04-05 15:43:04 last update 2014-04-05 15:45:32 | Answers (0) | 收藏

$n>2$ 维黎曼流形 $(M,g)$ 是 Einstein 流形当且仅当

$\text{Ric}=\frac{\text{scal}}{n}g.$

$\text{Ric}(v,w)=kg(v,w),\quad v,w\in T_p M.$

12. 流形上的 Laplace 算子

Posted by haifeng on 2014-04-05 11:10:52 last update 2014-08-20 11:12:31 | Answers (3) | 收藏

$\Delta f:=\text{div}(\text{grad}f).$

$\langle\text{grad}f, \xi\rangle=\xi f.$

$\text{div}X=\text{trace}(\xi\mapsto\nabla_{\xi}X).$

(1)

\begin{aligned} \text{div}(X+Y)&=\text{div}X+\text{div}Y,\\ \text{div}(fX)&=f(\text{div}X)+\langle\text{grad}f,X\rangle.\\ \end{aligned}

(2)

\begin{aligned} \Delta(f+h)&=\Delta f+\Delta h,\\ \text{div}(h(\text{grad}f))&=h(\Delta f)+\langle\text{grad}f,\text{grad}h\rangle,\\ \Delta(fh)&=h(\Delta f)+2\langle\text{grad}f,\text{grad}h\rangle+f(\Delta h).\\ \end{aligned}

$(\partial_j(p))f=\frac{\partial(f\circ x^{-1})}{\partial x^j}(x(p)).$

$\xi=\sum_{j=1}^{n}\xi^j\partial_j.$

$\xi f=\sum_j\xi^j\partial_j f.$

$\text{grad}(f)=\sum_{k,\ell}(g^{k\ell}\partial_{\ell}f)\partial_{k}.$

$\text{grad}f=\sum_{k,\ell}(g^{k\ell}\partial_\ell f)\partial_k,$

$\text{div}X=\frac{1}{\sqrt{\det(g_{ij})}}\sum_j\partial_j(X^j\sqrt{\det(g_{ij})}),$

$\Delta u=\frac{1}{\sqrt{\det(g_{ij})}}\partial_k\bigl(\sqrt{\det(g_{ij})}g^{k\ell}\partial_\ell u\bigr).$

References:

Isaac Chavel, Eigenvalues in Riemannian Geometry. Academic Press, 1984.

13. Sasaki 流形

Posted by haifeng on 2014-03-24 18:49:40 last update 2014-03-24 18:54:39 | Answers (0) | 收藏

[Def] $(S,g)$ 称为是 Sasaki 流形, 如果锥流形 $(C(S),\bar{g})=(\mathbb{R}_+\times S,dr^2+r^2g)$ 是 Kahler 流形.

$S$ 常等同于 $C(S)$ 的子流形 $\{1\}\times S$. 于是 $\dim S$ 是奇数, 不妨记为 $2m+1$, 而 $\dim_{\mathbb{C}}C(S)=m+1$.

$\xi:=J(\frac{\partial}{\partial r}),\quad\eta(Y):=g(\xi,Y),$

Prop. (a) $\xi$ 是 $S$ 上的一个 Killing 向量场. (b) $\xi$ 的积分曲线是一条测地线. (c) $\eta(\xi)=1$ 且 $d\eta(\xi,X)=0$, 对任意 $X\in\mathscr{X}(S)$.

Pf.

References: Akito Futaki, Hajime Ono, Guofang Wang, Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds, J. Differential Geometry, 83 (2009) 585--635.

14. Killing 向量场

Posted by haifeng on 2014-03-24 18:02:08 last update 2014-03-24 18:39:08 | Answers (0) | 收藏

$(L_X g)(Y,Z)=g(\nabla_Y X,Z)+g(Y,\nabla_Z X)=0,\quad\forall\ Y, Z\in\Gamma(TM)$

Pf.

$(L_X T)(Y_1,\ldots,Y_p)=D_X\bigl(T(Y_1,\ldots,Y_p)\bigr)-\sum_{i=1}^{p}T(Y_1,\ldots,L_X Y_i,\ldots,Y_p),$

$\begin{split} (L_X g)(Y,Z)&=Xg(Y,Z)-g(L_X Y,Z)-g(Y,L_X Z)\\ &=Xg(Y,Z)-g([X,Y],Z)-g(Y,[X,Z])\\ &=g(\nabla_X Y,Z)+g(Y,\nabla_X Z)-g([X,Y],Z)-g(Y,[X,Z])\\ &=g(\nabla_Y X,Z)+g(Y,\nabla_Z X). \end{split}$

$(L_X g)(Y,Z)=g(\nabla_Y X,Z)+g(Y,\nabla_Z X).$

15. 星算子

Posted by haifeng on 2014-03-22 11:23:57 last update 2014-03-22 11:24:20 | Answers (0) | 收藏

$A^p(V)$ 是由 $\eta_1\wedge\cdots\wedge\eta_p$ 等张成. $A(V)=\oplus_{p=0}^{n}A^p(V)$.

\begin{aligned} *(1)=\pm\zeta_1\wedge\cdots\wedge\zeta_n,\quad *(\zeta_1\wedge\cdots\wedge\zeta_n)=\pm 1\\ *(\zeta_1\wedge\cdots\wedge\zeta_p)=\pm\zeta_{p+1}\wedge\cdots\wedge\zeta_n. \end{aligned}

Exer1. 对于 $\omega\in A^p(V)$, 证明 $**\omega=(-1)^{p(n-p)}\omega$.

16. Ricci曲率正的紧致带边连通黎曼流形, 如果边界每一点处关于内法向量的平均曲率大于零, 则边界是连通的.

Posted by haifeng on 2014-03-02 14:34:45 last update 2014-03-02 16:26:36 | Answers (0) | 收藏

Thm. 设 $M$ 是一紧致连通的带边黎曼流形, $\text{Ric}(M)>0$. 设 $H_x$ 是边界 $\partial M$ 在 $x\in M$ 处关于内法向量的平均曲率, $H_x>0$, $\forall\ x\in\partial M$, 则 $\partial M$ 是连通的.

[分析] 利用弧长的第二变分公式可将 Ricci 曲率和平均曲率联系起来.

Reference:

H. Blaine Lawson, Jr. The Unknottedness of Minimal Embeddings. Invent. Math. 11. 183-187 (1970).

17. A conjecture of Gromov

Posted by haifeng on 2013-06-25 23:40:55 last update 2013-06-25 23:42:08 | Answers (0) | 收藏

$\text{vol}_{n}(\Omega)\leq\frac{\text{vol}_n(B^n(1))}{[\text{vol}_{n-1}(S^{n-1}(1))]^{\frac{n}{n-1}}}\cdot [\text{vol}_{n-1}(\partial\Omega)]^{\frac{n}{n-1}}$

Remark:

B. Kleiner 解决了 $n=3$ 的情形.

18. [open]非正截面曲率的曲面的球丛问题

Posted by haifeng on 2013-06-25 23:34:58 last update 2013-06-26 14:52:55 | Answers (0) | 收藏

$\text{vol}(\Omega)[\text{vol}(S\Sigma^2-\Omega)]=0 ?$

19. [open]Cheeger-Gromoll bundle

Posted by haifeng on 2013-06-25 23:25:58 last update 2013-06-25 23:25:58 | Answers (0) | 收藏

20. [open] Hopf 猜想

Posted by haifeng on 2013-06-25 23:23:13 last update 2013-06-25 23:23:13 | Answers (0) | 收藏

$S^2\times S^2$ 上是否存在这样的黎曼度量, 使得在该度量下有正的截面曲率?

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